Question

Which of the following functions represents a simple harmonic oscillation?

• A. $\sin \omega t -\cos \omega t$
• B. $\sin^2 \omega t$
• C. $\sin \omega t+\sin 2 \omega t$
• D. $\sin \omega t - \sin 2 \omega t$

Answer 1

Answer: (A)

$\sin \omega t -\cos \omega t$

Answer 2

One of the conditions for SHM is that restoring force $(F)$ and hence acceleration $(a)$ should be proportional to displacement $(y)$. Let, $y =\sin \omega t-\cos \omega t$ $\frac{d y}{d t} =\omega \cos \omega t+\omega \sin \omega t$ $\frac{d^{2} y}{d t^{2}} =-\omega^{2} \sin \omega t+\omega^{2} \cos \omega t$ or $a =-\omega^{2}(\sin \omega t-\cos \omega t)$ $a=-\omega^{2} y \Rightarrow a \propto-y$ This satisfies the condition of SHM. Other equations do not satisfy this condition.